In a first-order 2D translationally-active topological insulator, with band inversion at a non-\(\Gamma \)  point in the Brillouin zone (BZ), the defect effectively acts as an antiphase boundary (\(\Phi _{\rm dis}=\pi \) ) for the gapless helical edge modes. Consequently, it yields a Kramers pair of zero-energy bound states [1]}. In contrast, a second-order topological insulator features gapped edge states that stem from the mass domain wall in the bulk. But, the domain wall gaps out the edge only partially in turn producing the topological corner modes through the Jackiw-Rebbi mechanism [2]}. Consequently, the dislocation modes, originating from the gapped edge states across the antiphase boundary, as we show, still survive [Figs. REF (b),(c)], but are moved away from zero energy [Fig. REF (d)]. Importantly, when the orientation of the Burgers vector (\({\bf b}\) ) is parallel to the direction of the domain wall, the defect modes are pinned at zero energy. Hierarchy of HOT states in this way directly translates into the spectral flow of the dislocation modes, detectable in the tunneling spectroscopy measurements, for instance. Finally, the same mechanism is analogously operative for second-order three-dimensional (3D) translationally-active insulators: an edge dislocation hosts gapped modes which become gapless when the Burgers vector is parallel to the HOT mass domain wall, while a screw dislocation hosts gapless propagating modes only when it is orthogonal to a gapless surface [Fig. REF  and Fig. S1 of the SI].
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